Some aspects of quantum mechanics and field theory in a Lorentz invariant noncommutative space

TL;DR

This paper develops Feynman propagators and quantum field theory formulations for a Lorentz-invariant noncommutative spacetime based on the DFRA framework, exploring quantum mechanics, field theory, and potential early universe applications.

Contribution

It introduces a novel approach to noncommutative quantum mechanics and field theory within the DFRA framework, avoiding explicit NC parameters and enabling dimensional reduction to recover commutative spacetime.

Findings

Derived Feynman propagators for NC quantum mechanics in DFRA space

Analyzed NC free particle and harmonic oscillator systems

Extended formalism to NC scalar quantum field theory with interactions

Abstract

We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher-Fredenhagen-Roberts-Amorim (DFRA) NC background that can be considered as an alternative framework for the NC spacetime of the early Universe. The operators formalism was revisited and we applied its properties to obtain a NC transition amplitude representation. Two examples of DFRA's systems were discussed, namely, the NC free particle and NC harmonic oscillator. The spectral representation of the propagator gave us the NC wave function and energy spectrum. We calculated the partition function of the NC harmonic oscillator and the distribution function. Besides, the extension to NC DFRA quantum field theory is straightforward and we used it in a massive scalar field. We had written the scalar action with self-interaction $\phi^{4}$ using the Weyl-Moyal product to obtain the propagator and vertex of this model needed to perturbation theory. %and the effective action to give rise one-particle-irreducible diagrams. It is important to emphasize from the outset is that the formalism demonstrated here will not be constructed introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.